There is quite limited mathematical progress for the original nonlinear system since as pointed out by Kawashima in [1]}, the system \((\ref {chushifangcheng})\)  is neither symmetric hyperbolic nor strictly hyperbolic.
For this reason the classical local well-posedness theorem (cf. [2]}) cannot be directly applied to the system \((\ref {chushifangcheng})\) .
Moreover, the hydrodynamic and electrodynamic effects are strongly coupled.
The presence of strong nonlinearities and the lack of dissipation mechanism of the magnetic field also lead to many difficulties.
Due to the mathematically complicate structure of the original nonlinear system \((\ref {chushifangcheng})\) ,
some simplified models are derived according to the actual physical application.
As it was pointed out by Imai in [3]}, the assumption that the electric charge density \(\rho _{e} \simeq 0\)  is physically very good for the study of plasmas.
Here, it is essential to recognize that the quasi-neutrality assumption \(\rho _{e} \simeq 0\)  is different from the assumption of exact neutrality \(\rho _{e} = 0\)  since the latter would lead to the superfluous condition \(\operatorname{div}E=0 \) .
According to this quasi-neutrality assumption, we can eliminate the terms involving \(\rho _e\)  in the system \( (\ref {chushifangcheng}) \)  and derive the following simplified system (cf. [4]}):
\(\left\lbrace \begin{aligned}& \partial _{t} \rho +\operatorname{div}(\rho u)=0, \\[2mm]& \rho \left( \partial _{t}u + u \cdot \nabla u \right) + \nabla p = \mu ^{\prime } \Delta u + (\lambda +\mu ^{\prime }) \nabla (\operatorname{div} u) + J \times B, \\[2mm]& \frac{R}{\gamma -1} \rho (\partial _t \theta + u \cdot \nabla \theta ) + p \operatorname{div}u = \operatorname{div}(\kappa \nabla \theta ) + \mathbb {N}(u) + J \cdot (E + u\times B), \\[2mm]& \varepsilon \partial _{t} E - \frac{1}{\mu _0}\operatorname{curl}B+J=0, \\[2mm]& \partial _{t} B + \operatorname{curl} E=0, \quad \operatorname{div}B = 0,\end{aligned}\right.\) 
